The graded structure of \({B}({H}_4)\)  provided by the magnetic Fredholm module \((\rho ,{H}_4,F_{B,\varepsilon })\) 
and the involution \(\chi \)  is the starting point for the construction of a quantized differential calculus [1]}. In this calculus the role of the exterior derivative is played by the commutator with \(F_{B,\varepsilon }\) . This justifies the introduction of the following notation:
\(\,\mathrm {d}(T) := [F_{B,\varepsilon }, T]_\chi ,\qquad T\in {B}({H}_4) .\) 
